In the surreal and yet all-too-real opening scene of Emir Kusturica's 1995 film Underground, the Nazis bomb Belgrade zoo, causing the panicked animals to run for their lives. And it is during this same raid that the tiger of Téa Obreht's debut novel escapes to the hills above the fictional village of Galina.

This was 60 years ago, the narrator Natalia tells us, but the complicated story of what happened to the tiger and the people of Galina lives on. It's now rekindled by the death of Natalia's beloved grandfather. He was a native of Galina and just a boy when the tiger appeared. The key to her grandfather's life and death "lies between two stories: the story of the tiger's wife, and the story of the deathless man".

Natalia has followed in her grandfather's steps and become a doctor in "the City". On hearing of his death, she takes us on a labyrinthine journey to investigate. He has been her constant companion – their weekly visit to the city zoo was a ritual. A humanist schooled in the old tradition, he remained loyal to his patients even after he was expelled from the university for political reasons. He never parted with his copy of The Jungle Book, not even when a mysterious stranger dubbed "the deathless man" won it in a bet to prove his immortality to the rational doctor.

The deathless man is presented as a key piece in the puzzle, along with the bear-man, the tormented butcher-musician, his long-suffering and deaf Muslim wife who becomes the tiger's wife for reasons too complicated to explain here, and a whole menagerie of other rural Balkan curiosities whose stories are embroidered by a collective genius of superstition. The brilliant black comedy and matryoshka-style narrative are among the novel's great joys. But they are also one of the problems: after meeting innumerable exotic characters, it dawned on me that the back-stories stand in for a story, and style stands in for emotion.

Obreht's imagination is seductively extravagant and prone to folkloric hyperbole, and this makes parts of the novel read like a picaresque romp through some enchanted Balkan kingdom, rife with magic, murder and mayhem. Who cares, it's all a fable about a war – no, several wars – in some unnamed land. No real places or persons are named: Tito is "the Marshall", Belgrade is "the City", and we are in "a Balkan country still scarred by war".

But there is a sorrow that sometimes undercuts the flights of fancy, and this saves The Tiger's Wife from being a freak show. Obreht's – and Natalia's – real journey is back in time, and the real investigation here is of the difficult times, violent death and crippled afterlife of that mythical place once called Yugoslavia. The puzzle is Yugoslavia itself. The zoo was bombed again in Natalia's own time, the 1990s. The wolves ate their cubs, and the tiger, called Zbogom or Farewell, ate his own legs: a powerful metaphor for Serbia-Yugoslavia devouring her own children.

Téa Obreht was born in Belgrade in the 1980s, and her family left just as the war broke out, eventually settling in the US. This novel is a spirited attempt to cram her entire cultural and family heritage into a story. Modern Yugoslav fiction and film have tended towards the absurd, the hyperbolic and the surreal for a good reason: putting Yugoslavia's history into a coherent narrative is hard even for historians. Obreht has prodigious talent for storytelling and imagery, so it seems only a matter of time before she writes something truly great.

The Tiger's Wife is a frisky tiger cub chasing its tail – it covers a lot of ground, growls a lot, and never quite gets there, but we have fun along the way. What the novel lacks in emotional depth, it makes up for in personality and sheer wackiness. But the real delirium – and the real emotion – doesn't lie in the stories of tiger-men, bear-men, deathless men and their consorts. It's found in the students of medicine in the war-stricken "City" trying to get skulls on the black market. It's there when the grandfather is tipped off by Serbian mercenaries about their plan to bomb the Muslim-dominated town of Sarobor (read Mostar of the destroyed bridge), and instead of getting out of there fast, he goes to an old restaurant, where a poignantly polite waiter serves him a last feast. He remembers his life here with his Bosnian wife, he bids farewell to the old Yugoslavia, and muses that "my name, your name, her name. In the end, all you want is someone to long for you when it comes time to put you in the ground."

Then the sky turns red with fire and the real law of the jungle comes to town.

Kapka Kassabova's novel Villa Pacifica will be published by Alma Books in August.